Question bank IMO Shortlist 1966

(USS, 1966) Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac16 S$.

(USS, 1966) Find the last two digits of a sum of eighth powers of $100$ consecutive integers.

(USS, 1966) Given the vertex $A$ and the centroid $M$ of a triangle $ABC$, find the locus of vertices $B$ such that all the angles of the triangle lie in the interval $[40^\circ,70^\circ]$.

(USS, 1966) Let $ABCD$ be a tetrahedron such that $AB\perp CD$, $AC\perp BD$, and $AD\perp BC$. Prove that the midpoints of the edges of the tetrahedron lie on a sphere.

(USS, 1966) Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

(CZS) We are given $n>3$ points in the plane, no three of which lie on a line. Does there necessarily exist a circle that passes through at least three of the given points and contains none of the other given points in its interior?

(GDR) Given $n$ positive real numbers $a_1,a_2,\ldots,a_n$ such that $a_1a_2\cdots a_n=1$, prove that
$$(1+a_1)(1+a_2)\cdots(1+a_n)\ge2^n.$$

(BUL) A regular triangular prism has height $h$ and a base of side length $a$. Both bases have small holes in the centers, and the inside of the three vertical walls has a mirror surface. Light enters through the small hole in the top base, strikes each vertical wall once and leaves through the hole in the bottom. Find the angle at which the light enters and the length of its path inside the prism.

(POL) Five points in the plane are given, no three of which are collinear. Show that some four of them form a convex quadrilateral.

(USS) Prove the inequality
$$\tan\frac{\pi\sin x}{4\sin\alpha}+\tan\frac{\pi\cos x}{4\cos\alpha}>1$$
for any $x,\alpha$ with $0\le x\le\pi/2$ and $\pi/6\lt y\lt \pi/3$.

(USS) A convex planar polygon $\mathcal M$ with perimeter $l$ and area $S$ is given. Let $M(R)$ be the set of all points in space that lie a distance at most $R$ from a point of $\mathcal M$. Show that the volume $V(R)$ of this set equals
$$V(R)=\frac43\pi R^3+\frac\pi2 lR^2+2SR.$$

(USS) For which arrangements of two infinite circular cylinders does their intersection lie in a plane?

(USS) We are given a bag of sugar, a two-pan balance, and a weight of 1 gram. How do we obtain 1 kilogram of sugar in the smallest possible number of weighings?

(ROM) Find $x$ such that
$$\frac{\sin3x\cos(60^\circ-4x)+1}{\sin(60^\circ-7x)-\cos(30^\circ+x)+m}=0,$$
where $m$ is a fixed real number.

(GDR) How many real solutions are there to the equation $x=1964\sin x-189$?

(CZS) Does there exist an integer $z$ that can be written in two different ways as $z=x!+y!$, where $x,y$ are natural numbers with $x\le y$?

(BUL) Find digits $x,y,z$ such that the equality $$\sqrt{\underbrace{\overline{xx\cdots x}}_{2n}-\underbrace{\overline{yy\cdots y}}_n}=\underbrace{\overline{zz\cdots z}}_n$$ holds for at least two values of $n\in\mathbb N$, and in that case find all $n$ for which this equality is true.

(YUG) Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Prove the inequality $$\binom n2\sum_{i<j}\frac1{a_i a_j}\ge4\left(\sum_{i<j}\frac1{a_i+a_j}\right)^2$$ and find the conditions on the numbers $a_i$ for equality to hold.

(POL) Compute the largest number of regions into which one can divide a disk by joining $n$ points on its circumference.

(POL) Points $A,B,C,D$ lie on a circle such that $AB$ is a diameter and $CD$ is not. If the tangents at $C$ and $D$ meet at $P$ while $AC$ and $BD$ meet at $Q$, show that $PQ$ is perpendicular to $AB$.